The way of making sound on the strings of a bowed string instrument by rubbing them with a bow made from horsehair appears to me as something so particular that it must have been very hard and unlikely to come across this idea. How did such a combination of a way to make sound and of a particular material, happen to be discovered?

Horsehair was not always an unusual material....
– David BowlingJan 10 at 22:21

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Horsehair has always been perfect. Ever petted a horses mane? The hairs of the mane and tail have many barbs on them that make them directionally “grabby”. The tiny barbs grasp the catgut strings, then releases them due to the bowing motion, causing a sawtooth type wave to be imparted onto the string, which travels up and down the string from the bridge to the nut or the fingers... this fact was not lost to tribes who’s livelihood depended on horses.
– Richard BarberJan 11 at 5:39

Are all the various bowed instruments of the world, played with horsehair? A very quick wiki search of Africa and China instruments turned up only horsehair.
– Michael CurtisJan 11 at 19:25

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There are only two possible kind of vibrations with a string fixed at both ends:

Plucking; the vertical impulse leads to a transversal vibration

twisting, leading to a torsional vibration

To get the string to twist, rosin is applied to the horse hair so that it can grip the string. You need something soft as an attempt with a stick will show and also some tension behind it, so more force may achieve a louder sound.

Rubbing is no completely exotic method for sound generation as can be heard by rubbing a wet finger on glass (finally leading to glassharmonica). You are right, that it is somewhat non-obvious, but that's a reason, why string instruments with bow or wheel (hurdy-gurdy) appeared not before tenth century and so are comparatively young especially compared to wind instruments, where a bone with a hole is a sufficient prototype.

English wikipedia cites the 1988 Encyclopedia Britannica with this (in line with the New Grove Dictionary:

… bowing can be traced as far back as the Islamic civilization of the 10th century … it seems likely that the principle of bowing originated among the nomadic horse riding cultures of Central Asia …

This would make horse hair the first choice.

Just to round up: one of the oldest bowed string Instruments is the rebec and some nice historical representations are collected here. It closes with a literature list containing this for further reading:

Bachmann, Werner. The Origins of Bowing. trans Norma Deane. Oxford University Press: London, 1969. This is the root source for most modern research, and practically everybody cites him. A very good comprehensive study of early string and bowed string instruments.

This answers part of it - but it would be interesting to track down any history of experimentation with different materials - maybe starting with a bone or tree branch rubbed across the strings, leading to trials with different stranded material and so on.
– Carl WitthoftJan 11 at 13:49

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NitPick: you can excite a transverse wave as well. Consider the extreme case of a "Slinky" , fixed at both ends. It's easy to "pluck" it longitudinally and watch the wave propogate. In fact, a similar coiled metal under slight tension will propogate a sound wave slowly enough to produce an "echo" effect!
– Carl WitthoftJan 11 at 13:53